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2014 | OriginalPaper | Chapter

Mandelbrot Cascades and Related Topics

Author : Julien Barral

Published in: Geometry and Analysis of Fractals

Publisher: Springer Berlin Heidelberg

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Abstract

This article is an extended version of the talk given by the author at the conference “Advances in fractals and related topics”, in December 2012 at the Chinese Hong-Kong University. It gathers recent advances in Mandelbrot cascades theory and related topics, namely branching random walks, directed polymers on disordered trees, multifractal analysis, and dynamical systems.

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next chapter Law of Pure Types and Some Exotic Spectra of Fractal Spectral Measures
Appendix
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Literature
[Aid13]
[AS14]
Aidekon, E., Shi, Z.: The Seneta-Heyde scaling for the branching random walk. Ann. Probab. 42, 959–993 (2014)
[ABM12]
Alsmeyer, G., Biggins, J.-D., Meiners, M.: The functional equation of the smoothing transformation. Ann. Probab. 140, 2069–2105 (2012)MathSciNetCrossRef
[AM12]
[AM13]
Alsmeyer, G., Meiners, M.: Fixed points of the smoothing transformation: two-sided case. Probab. Theory Relat. Fields 155, 165–199 (2013)MathSciNetMATHCrossRef
[AJKS11]
[AB14]
Attia, N., Barral, J.: Hausdorff and packing spectra, large deviations, and free energy for branching random walks. in \({\mathbb{R}}^d\). To appear in Comm. Math. Phys (2014). arXiv:​1305.​2034
[BM03]
[Bar00]
Barral, J.: Continuity of the multifractal spectrum of a random statistically self-similar measure. J. Theor. Probab. 13, 1027–1060 (2000)MathSciNetMATHCrossRef
[BBP03]
Barral, J., Ben Nasr, F., Peyrière, J.: Comparing multifractal formalisms: the neighboring boxes conditions. Asian J. Math. 7, 149–166 (2003)MathSciNetMATH
[BF05]
[BFP10]
Barral, J., Fan, A.-H., Peyrière, J.: Mesures engendrées par multiplications. In: Quelques Interactions Entre Analyse, Probabilités et Fractals, Panoramas et Synthéses, vol. 32, Soc. Math., France (2010)
[BJ10]
[BJM10]
Barral, J., Jin, X., Mandelbrot, B.B.: Uniform convergence for complex \([0,1]\)-martingales. Ann. Appl. Probab. 20, 1205–1218 (2010)MathSciNetMATHCrossRef
[BJM10a]
Barral, J., Jin, X., Mandelbrot, B.B.: Convergence of signed multiplicative cascades. Ann. Appl. Probab. 20, 1219–1252 (2010)
[BJRV13]
Barral, J., Jin, X., Rhodes, R., Vargas, V.: Gaussian multiplicative chaos and KPZ duality. Commun. Math. Phys. 323, 451–485 (2013)
[BKNSW14a]
Barral, J., Kupiainen, A., Nikula, M., Saksman, E., Webb, C.: Critical Mandelbrot cascades. Commun. Math. Phys. 325, 685–711 (2014)
[BKNSW14+]
Barral, J., Kupiainen, A., Nikula, M., Saksman, E., Webb, C.: Basic properties of critical lognormal multiplicative chaos. To appear in Ann. Probab. arXiv:​1303.​4548
[BM02]
[BM04]
Barral, J., Mandelbrot, B.: Introduction to infinite products of random independent functions (Random multiplicative multifractal measures, Part I). In: Lapidus, M.L., van Frankenhuysen, M. (eds.) Fractal Geometry and Applications: A Jubilee of Benot Mandelbrot. Proceedings of Symposia in Pure Mathematics, vol. 72, Part 2, pp. 3–16. AMS, Providence, RI (2004)
[BM04a]
Barral, J., Mandelbrot, B.: Non-degeneracy, moments, dimension, and multifractal analysis for random multiplicative measures (Random multiplicative multifractal measures, Part II), pp. 17–52. In: Proceedings of Symposia in Pure Mathematics, vol. 72, Part 2. AMS, Providence, RI (2004)
[BM09]
Barral, J., Mandelbrot, B.B.: Fractional multiplicative processes. Ann. Inst. Henry Poincaré Probab. Stat. 45, 1116–1129 (2009)
[BPW09]
Barral, J., Peyrière, J., Wen, Z.: Dynamics of Mandelbrot cascades. Probab. Theory Relat. Fields 144, 615–631 (2009)MATHCrossRef
[BRV12]
Barral, J., Rhodes, R., Vargas, V.: Limiting laws of supercritical branching random walks, C. R. Acad. Sci. Paris Ser. I 350, 535–538 (2012)
[BS05]
Barral, J., Seuret, S.: From multifractal measures to multifractal wavelet series. J. Fourier Anal. Appl. 11, 589–614 (2005)
[BS07]
Barral, J., Seuret, S.: Heterogeneous ubiquitous systems in \(\mathbb{R}^d\) and Hausdorff dimension. Bull. Braz. Math. Soc. 38, 467–515 (2007)MathSciNetMATHCrossRef
[BS07]
[BS09]
Benjamini, I., Schramm, O.: KPZ in one dimensional random geometry of multiplicative cascades. Commun. Math. Phys. 289, 653–662 (2009)MathSciNetMATHCrossRef
[Ber96]
Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996)
[Big77]
[Big79]
[Big92]
[BK97]
[BK04]
[BK05]
Biggins, J.D., Kyprianou, A.: The smoothing transform: the boundary case. Electron. J. Probab. 10, 609–631 (2005)MathSciNetCrossRef
[BHJ11]
Biggins, J.D., Hambly, B.M., Jones, O.D.: Multifractal spectra for random self-similar measures via branching processes. Adv. Appl. Probab. 43, 1–39 (2011)MathSciNetMATHCrossRef
[Bov06]
Bovier, A.: Statistical mechanics of disordered system. A mathematical perspective. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 18. Cambridge University Press, Cambridge (2006)
[Bra83]
Bramson, M.: Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Am. Math. Soc. 44(285), iv\(+\)190 pp. (1983)
[Bur09]
Buraczewski, D.: On tails of fixed points of the smoothing transform in the boundary case. Stoch. Process. Appl. 119(11), 3955–3961 (2009)MathSciNetMATHCrossRef
[CD01]
Carpentier, D., Le Doussal, P.: Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and Sinh-Gordon models. Phys. Rev. E 63, 026110 (2001)CrossRef
[CRV13]
Chevillard, L., Rhodes, R., Vargas, V.: Gaussian multiplicative chaos for symmetric isotropic matrices. J. Stats. Phys. 150, 678–703 (2013)
[CK92]
[DZ98]
Dembo, A., Zeitouni, O.: Large deviations techniques and applications. In: Applications of Mathematics, vol. 38, 2nd edn. Springer, New York (1998)
[DS88]
[DES93]
Derrida, B., Evans, M.R., Speer, E.R.: Mean field theory of directed polymers with random complex weights. Commun. Math. Phys. 156(2), 221–244 (1993)MathSciNetMATHCrossRef
[DMPV95]
Dodson, M.M., Melián, M.V., Pestana, D., Vélani, S.L.: Patterson measure and ubiquity. Ann. Acad. Sci. Fenn. Ser. A I Math. 20 37–60 (1995)
[DS09]
[DS11]
[DRSV00a]
Duplantier, B., Rhodes, R., Sheffield, S., Vargas, V.: Critical Gaussian multiplicative chaos: convergence of the derivative martingale. arXiv:​1206.​1671
[DRSV00]
Duplantier, B., Rhodes, R., Sheffield, S., Vargas, V.: Renormalization of critical Gaussian multiplicative chaos and KPZ formula. arXiv:​1212.​0529
[DL83]
[Fal94]
[Fal03]
Falconer, K.J.: Fractal Geometry. Mathematical Foundations and Applications, 2nd edn. Wiley, New York (2003)
[Fan89]
Fan, A.H.: Chaos additifs et chaos multiplicatifs de Lévy. C. R. Acad. Sci. Paris Sér. I 308, 151–154 (1989)
[Fan97]
Fan, A.H.: Sur le chaos de Lévy d’indice \(0<\alpha <1\). Ann. Sci. Math. Que. 21, 53–66 (1997)
[Fan04]
Fan, A.-H.: Some topics in the theory of multiplicative chaos. In: Bandt, Ch., Zaehle, M., Mosco, U. (eds.) Progress in Probability, vol. 57, pp. 119–134. Birkhäuser, Basel (2004)
[Fra95]
Franchi, J.: Chaos multiplicatif: un traitement simple et complet de la fonction de partition. Séminaire Probab. Strasbourg 29, 194–201 (1995)MathSciNet
[FP88]
Frisch, U., Parisi, G.: Fully developed turbulence and intermittency in turbulence. Proceedings of the International School of Physics “Enrico Fermi”. In: Ghil, M. (ed.) Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, course 88, pp. 84–88. North Holland, Amsterdam (1985)
[Gui90]
Guivarch, Y.: Sur une extension de la notion de loi semi-stable. Ann. Inst. Henry Poincaré Probab. Stat. 26, 261–285 (1990)MathSciNet
[HW92]
Holley, R., Waymire, E.C.: Multifractal dimensions and scaling exponents for strongly bounded random fractals. Ann. Appl. Probab. 2, 819–845 (1992)MathSciNetMATHCrossRef
[HS09]
Hu, Y., Shi, Z.: Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37(2), 742–789 (2009)MathSciNetMATHCrossRef
[Jaf98]
Jaffard, S.: Oscillations spaces: properties and applications to fractal and multifractal functions. J. Math. Phys. 39(8), 4129–4141 (1998)MathSciNetMATHCrossRef
[Jaff99]
[Jaf00a]
[Jaf00]
[Jaf04]
Jaffard, S.: Wavelets techniques in multifractal analysis. In: Lapidus, M., Frankenhuijsen, M.V. (eds.) Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot. Proceedings of Symposia in Pure Mathematics, vol. 72(2), pp. 91–151 (2004)
[Jin11]
[Jin12]
[Jin14]
Jin, X.: A uniform result for two dimensional fractional multiplicative processes. Ann. Inst. H. Poincaré Probab. Stat. 50(2), 512–523 (2014)
[JW11]
Johnson, T., Waymire, E.: Tree polymers in the infinite volume limit at critical strong disorder. J. Appl. Probab. 48, 885–891 (2011)MathSciNetMATHCrossRef
[Kah74]
Kahane, J.-P.: Sur le modele de turbulence de Benoit Mandelbrot. C. R. Acad. Sci. Paris, 278, 567–569 (1974)
[Kah85]
[Kah87a]
Kahane, J.-P.: Positive martingales and random measures. Chi. Ann. Math. 8B1, 1–12 (1987)
[KP76]
Kahane, J.-P., Peyrière, J.: Sur certaines martingales de B. Mandelbrot Adv. Math. 22, 131–145 (1976)MATHCrossRef
[Kah87]
Kahane, J.-P.: Multiplications aléatoires et dimensions de Hausdorff. Ann. Inst. Henri Poincaré Probab. Stat. 23, 289–296 (1987)MathSciNetMATH
[Kah91]
Kahane, J.-P.: Produits de poids aléatoires indépendants et applications. In: Fractal Geometry and Analysis (Montreal, PQ, 1989), pp. 277–324. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 346. Kluwer, Dordrecht (1991)
[KPZ98]
Knizhnik, V.G., Polyakov, A.M., Zamolodchikov, A.B.: Fractal structure of 2D-quantum gravity. Mod. Phys. Lett A 3(8), 819–826 (1988)
[Kol62]
Kolmogorov, A.N.: Précisions sur la structure locale de la turbulence dans un fluide visqueux aux nombres de Reynolds élevés, Mécanique de la turbulence, Colloq. Intern. CNRS, Marseille (1961) [Editions CNRS, pp. 447–451 (1962)]
[LRV00]
[Liu98]
Liu, Q.: Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. Appl. Probab. 30, 85–112 (1998)MATHCrossRef
[Liu00]
Liu, Q.: On generalized multiplicative cascades. Stoch. Process. Appl. 86(2), 263–286 (2000)MATHCrossRef
[Liu01]
Liu, Q.: Asymptotic properties and absolute continuity of laws stable by random weightedmean. Stoch. Proc. Appl. 95, 83–107 (2001)MATHCrossRef
[Mad00]
[Man72]
Mandelbrot, B.B.: Possible refinement of the lognormal hypothesis concerning the distribution of energy in intermittent turbulence. Statistical models and turbulence. In: Rosenblatt, M., Atta, C.V. (eds.) Lectures Notes in Physics, vol. 12, 333–351. Springer, New York (1972)
[Man74a]
Mandelbrot, B.B.: Multiplications aléatoires itérées et distributions invariantes par moyennes pondérées. C. R. Acad. Sci. Paris 278, 289–292 and 355–358 (1974)
[Man74b]
Mandelbrot, B.B.: Intermittent turbulence in self-similar cascades, divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331–358 (1974)MATHCrossRef
[Mat95]
Mattila, P.: Geometry of sets and measures in Euclidean spaces, fractals and rectifiability. In: Cambridges Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995)
[Mol96]
Molchan, G.M.: Scaling exponents and multifractal dimensions for independent random cascades. Commun. Math. Phys. 179(3), 681–702 (1996)MathSciNetMATHCrossRef
[Ols94]
Olsen, L.: Random geometrically graph directed self-similar multifractals, Pitman Res. Notes Math. Ser. 307 (1994)
[Ols95]
[OW00]
Ossiander M., Waymire, E.C.: Statistical estimation for multiplicative cascades. Ann. Stat. 28, 1–29 (2000)
[Pey74]
Peyrière, J.: Turbulence et dimension de Hausdorff. C. R. Acad. Sci. Paris 278, 567–569 (1974)MATH
[Pey77]
[Pey79]
Peyrière, J.: A singular random measure generated by spliting \([0,1]\). Z. Wahrsch. Verw. Geb. 47, 289–297 (1979)MATHCrossRef
[Pey00]
Peyrière, J.: Recent results on Mandelbrot multiplicative cascades. In: Bandt, Ch., Graf, S., Zähle, M. (eds.) Fractal Geometry and Stochastics II, pp. 147–159. Progress in Probability, vol. 46. Birkhäuser, Basel (2000)
[RV10]
[RV08]
[RV13]
[She00]
[She72]
[WW69]
Waymire, E.C., Williams, S.C.: A cascade decomposition theory with applications to Markov and exchangeable cascades. Trans. Am. Math. Soc. 348, 585–632 (1996)MathSciNetMATHCrossRef
[WW13]
Waymire, E.C., Williams, S.C.: \(T\)-martingales, size-biasing and tree polymer cascades. In: Barral, J., Seuret, S. (eds.) Further Developments in Fractals and Related Fields. Birkhäuser, Basel (2013)
[Web11]
Webb, C.: Exact asymptotics of the freezing transitions of a logarithmically correlated random energy model. J. Stat. Phys. 145, 1595–1619 (2011)MathSciNetMATHCrossRef
Metadata
Title
Mandelbrot Cascades and Related Topics
Author
Julien Barral
Copyright Year
2014
Publisher
Springer Berlin Heidelberg
Book
Geometry and Analysis of Fractals

Print ISBN: 978-3-662-43919-7

Electronic ISBN: 978-3-662-43920-3

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-3-662-43920-3_1

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